tex/computing-solutions.tex

@@ -13,7 +13,7 @@ a different algorithm will be presented making use of theorems from Section
 with the goal of cutting down the run time.
 
 \subsubsection{Finding possible \texorpdfstring{$r$}{r} and
-\texorpdfstring{$c$}{c}}
+	\texorpdfstring{$c$}{c}}
 To do this, first calculate the upper bound $r_{\mathrm{max}}$ on the ranks of tilt
 semistabilisers, as given by Theorem \ref{thm:loose-bound-on-r}.
 
@@ -36,7 +36,7 @@ the Bogomolov inequalities and Consequence 3 of Lemma
 ($\chern_2^{\beta_{-}}(u)>0$).
 
 \subsubsection{Finding \texorpdfstring{$d$}{d} for fixed \texorpdfstring{$r$}{r}
-and \texorpdfstring{$c$}{c}}
+	and \texorpdfstring{$c$}{c}}
 
 $\Delta(u) \geq 0$ induces an upper bound $\frac{c^2}{2r}$ on $d$, and the
 $\chern_2^{\beta_{-}}(u)>0$ condition induces a lower bound on $d$.
@@ -51,8 +51,8 @@ end up not yielding any solutions for the problem.
 In fact, solutions $u$ to our problem with high rank must have $\mu(u)$ close to
 $\beta_{-}(v)$:
 \begin{align*}
-	0 &\leq \chern_1^{\beta_{-}}(u) \leq \chern_1^{\beta_{-}}(u) \\
-	0 &\leq \mu(u) - \beta_{-} \leq \frac{\chern_1^{\beta_{-}}(v)}{r}
+	0 & \leq \chern_1^{\beta_{-}}(u) \leq \chern_1^{\beta_{-}}(u)      \\
+	0 & \leq \mu(u) - \beta_{-} \leq \frac{\chern_1^{\beta_{-}}(v)}{r}
 \end{align*}
 In particular, it is the $\chern_1^{\beta_{-}}(v-u) \geq 0$ condition which
 fails for $r$,$c$ pairs with large $r$ and $\frac{c}{r}$ too far from $\beta_{-}$.
@@ -66,17 +66,17 @@ alternative algorithm which will later be described in Section
 \ref{sect:prob2-algorithm}.
 
 \begin{center}
-\label{table:bench-schmidt-vs-nay}
-\begin{tabular}{ |r|l|l| }
- \hline
- Choice of $v$ on $\mathbb{P}^2$
- & $(3, 2\ell, -2)$
- & $(3, 2\ell, -\frac{15}{2})$ \\
- \hline
- \cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s & >1hr \\
- \cite{NaylorRust2023} exec time & \sim 50ms & \sim 50ms \\
- \hline
-\end{tabular}
+	\label{table:bench-schmidt-vs-nay}
+	\begin{tabular}{ |r|l|l| }
+		\hline
+		Choice of $v$ on $\mathbb{P}^2$
+		                                                             & $(3, 2\ell, -2)$
+		                                                             & $(3, 2\ell, -\frac{15}{2})$             \\
+		\hline
+		\cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s                    & >1hr      \\
+		\cite{NaylorRust2023} exec time                              & \sim 50ms                   & \sim 50ms \\
+		\hline
+	\end{tabular}
 \end{center}
 
 \section{Computing Solutions to Problem \ref{problem:problem-statement-2}}
@@ -94,7 +94,7 @@ The algorithm yields solutions
 $u=(r,c\ell,d\ell^2)$ to the problem as follows.
 
 \subsubsection{Iterating Over Possible
-\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}}
+	\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}}
 
 Given a Chern character $v$, the domain of the problem are first verified: that
 $v$ has positive rank, that it satisfies $\Delta(v) \geq 0$, and that
@@ -102,6 +102,7 @@ $\beta_{-}(v)$ is rational.
 Take $\beta_{-}(v)=\frac{a_v}{n}$ in simplest terms.
 Iterate over $q = \frac{b_q}{n} \in (0,\chern_1^{\beta_{-}}(v))\cap\frac{1}{n}\ZZ$.
 The code used to generate the corresponding values for $b_q$ is shown in Listing
+% texlab: ignore
 \ref{fig:code:consideredb}.
 
 \lstinputlisting[
@@ -118,6 +119,7 @@ We can therefore reduce the problem of finding solutions to the more specialised
 problem of finding the solutions $u$ with each fixed possible $q=\chern_1^\beta(u)$
 (i.e. choice of $b$).
 The code representing this appears in Listing
+% texlab: ignore
 \ref{fig:code:reducingtoeachb}.
 Line 16 refers to creating an objects representing the context the specialised
 problem for the fixed $q$ value, with the next line `solving' the specialised
@@ -133,9 +135,9 @@ and collect up the results.
 ]{../tilt.rs/src/tilt_stability/find_all.git-untrack.rs.tex.git-untrack}
 
 \subsubsection{Iterating Over Possible
-\texorpdfstring{$r=\chern_0(u)$}{r}
-for Fixed
-\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
+	\texorpdfstring{$r=\chern_0(u)$}{r}
+	for Fixed
+	\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
 }
 
 Let $q=\frac{b_q}{n}$, for which we are now solving the more specialised problem of finding
@@ -159,6 +161,7 @@ Fixing $r$ and $q$ also determines $c\coloneqq\chern_1(u)$, and so we can genera
 the corresponding values of $c$, as we generate the $r$ values.
 It now remains to solve the problem for each of the combinations of fixed values
 for $q$ and $r$ (and consequently $c$) considered.
+% texlab: ignore
 This is shown in Listing \ref{fig:code:reducingtoeachr}.
 
 \lstinputlisting[
@@ -170,11 +173,11 @@ This is shown in Listing \ref{fig:code:reducingtoeachr}.
 
 
 \subsubsection{Iterating Over Possible
-\texorpdfstring{$d=\chern_2(u)/\ell^2$}{d}
-for Fixed
-\texorpdfstring{$r=\chern_0(u)$}{r}
-and
-\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
+	\texorpdfstring{$d=\chern_2(u)/\ell^2$}{d}
+	for Fixed
+	\texorpdfstring{$r=\chern_0(u)$}{r}
+	and
+	\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
 }
 
 At this point we are considering a specialisation of the problem
@@ -198,8 +201,10 @@ equivalent to bounds on $d$ given by the equations
 in Subsubsection \ref{subsubsect:all-bounds-on-d-prob2}
 It therefore remains to just pick values
 $d\in\frac{1}{\lcm(m,2n^2)}\ZZ$ within the bounds.
+% texlab: ignore
 Listing \ref{fig:code:solveforfixedr} is the code for solving this
 specialisation of the problem, where the possible $d$ values are computed in
+% texlab: ignore
 Listing \ref{fig:code:possible_chern2}.
 The explicit code for the bounds can be found in Appendix
 \ref{appendix:subsubsec:fixed-r}.
@@ -232,11 +237,11 @@ decrease in computational time to find the solutions to the problem.
 This could be due to a range of potential reasons:
 \begin{itemize}
 	\item Unexpected optimisations from the compiler for a certain form of the
-		program.
+	      program.
 	\item Increased complexity to computing the formulae for the tighter bounds.
 	\item Modern CPU architecture such as branch predictors
-		\cite{BranchPredictor2024} may offset the overhead of considering ranks that
-		turn out to be too large to have any solutions.
+	      \cite{BranchPredictor2024} may offset the overhead of considering ranks that
+	      turn out to be too large to have any solutions.
 \end{itemize}
 
 For relatively small Chern characters (as those appearing in examples so far),